The interesting part of the cohomology of the theta divisor $D$ of an abelian fivefold $A$ shares numerical properties with the Lie algebra $E_6$. We define 27 surfaces inside $D$, one for each realisation of $A$ as a Prym variety, and explain how they generate a sublattice of $H^4(D, \mathbb{Z})$ isomorphic to the root lattice of $E_6$. This gives an effective proof of the Hodge conjecture for the theta divisor.